Function holomorphic at a point - Revision history

Vipul: 2 revisions

2008-05-18T19:12:41Z

2 revisions

← Older revision	Revision as of 19:12, 18 May 2008
(No difference)

Vipul at 21:19, 16 April 2008

2008-04-16T21:19:59Z

← Older revision		Revision as of 21:19, 16 April 2008
Line 6:		Line 6:

	Let <math>U</math> be an open subset (without loss of generality, an open connected subset, i.e. a [[domain]]) in <math>\mathbb{C}</math>. Let <math>f:U \to \mathbb{C}</math> be a function, and let <math>z_0 \in U</math>. We say that <math>f</math> is '''holomorphic''' at <math>z_0</math> if there exists an open subset <math>V \ni z_0</math> such that <math>f</math> is [[function complex-differentiable at a point\|complex-differentiable]] for every <math>z \in V</math>, and further, if the complex differential <math>f'</math> is a continuous function on <math>V</math>.		Let <math>U</math> be an open subset (without loss of generality, an open connected subset, i.e. a [[domain]]) in <math>\mathbb{C}</math>. Let <math>f:U \to \mathbb{C}</math> be a function, and let <math>z_0 \in U</math>. We say that <math>f</math> is '''holomorphic''' at <math>z_0</math> if there exists an open subset <math>V \ni z_0</math> such that <math>f</math> is [[function complex-differentiable at a point\|complex-differentiable]] for every <math>z \in V</math>, and further, if the complex differential <math>f'</math> is a continuous function on <math>V</math>.

			It turns out that this is equivalent to a [[function complex-analytic at a point]].

Vipul: New page: {{at-point function property|holomorphic function}} ==Definition== ===In one dimension=== Let $U$ be an open subset (without loss of generality, an open connected subset, i.e...

2008-04-16T21:13:42Z

New page: {{at-point function property|holomorphic function}} ==Definition== ===In one dimension=== Let <math>U</math> be an open subset (without loss of generality, an open connected subset, i.e...

New page

{{at-point function property|holomorphic function}}

==Definition==

===In one dimension===

Let <math>U</math> be an open subset (without loss of generality, an open connected subset, i.e. a [[domain]]) in <math>\mathbb{C}</math>. Let <math>f:U \to \mathbb{C}</math> be a function, and let <math>z_0 \in U</math>. We say that <math>f</math> is '''holomorphic''' at <math>z_0</math> if there exists an open subset <math>V \ni z_0</math> such that <math>f</math> is [[function complex-differentiable at a point|complex-differentiable]] for every <math>z \in V</math>, and further, if the complex differential <math>f'</math> is a continuous function on <math>V</math>.

Function holomorphic at a point - Revision history

Vipul: 2 revisions

Vipul at 21:19, 16 April 2008

Vipul: New page: {{at-point function property|holomorphic function}} ==Definition== ===In one dimension=== Let U be an open subset (without loss of generality, an open connected subset, i.e...

Vipul: New page: {{at-point function property|holomorphic function}} ==Definition== ===In one dimension=== Let $U$ be an open subset (without loss of generality, an open connected subset, i.e...